"Root systems and Dynkin diagrams(mathematica)"의 두 판 사이의 차이

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* Root Systems and Dynkin diagrams<br>     * http://en.wikipedia.org/wiki/root_systems<br>     * http://en.wikipedia.org/wiki/Dynkin_diagram<br><br><br><br> A_n root systems<br><br> (* A_n type Cartan matrix *)<br> r := 3<br> rt[i_] := UnitVector[r + 1, i] - UnitVector[r + 1, i + 1]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br><br> B_n root systems<br><br> Clear[rt]<br> (*B_r type Cartan matrix*)<br> r := 4<br> rt[i_] :=<br>  If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i], UnitVector[r, r]]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br><br> C_n root systems<br><br> Clear[rt]<br> (*C_r type Cartan matrix*)<br> r := 4<br> rt[i_] :=<br>  If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i], 2*UnitVector[r, r]]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br><br> D_n root systems<br><br> Clear[rt]<br> (*D_r type Cartan matrix*)<br> r := 6<br> rt[i_] :=<br>  If[i < r, UnitVector[r, i] - UnitVector[r, 1 + i],<br>   UnitVector[r, r - 1] + UnitVector[r, r]]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> CA := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> CA // MatrixForm<br><br><br><br> G2 root system<br><br> Clear[r, rt]<br> (*G_ 2 type Cartan matrix*)<br> r := 2<br> rt[1] := {1, -1, 0}<br> rt[2] := {-1, 2, -1}<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> CA := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> A // MatrixForm<br> rt[1]<br> rt[2] - rt[1]<br><br><br><br> F4 root system<br><br> Clear[r, rt]<br> (*F_ 4 type Cartan matrix*)<br> Clear[rt]<br> r := 4<br> rt[1] := {1, -1, 0, 0}<br> rt[2] := {0, 1, -1, 0}<br> rt[3] := {0, 0, 1, 0}<br> rt[4] := {-1, -1, -1, -1}/2<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}]<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br><br><br> E_6 root system<br><br> Clear[rt]<br> (*E_ 6 type Cartan matrix*)<br> r := 6<br> rt[1] := {0, 0, 0, 0, 0, 0, 0, 1, \[Minus]1}<br> rt[2] := {0, 0, 0, 0, 0, 0, 1, -1, 0}<br> rt[3] := {1, -2, 1, -2, 1, 1, -2, 1, 1}/3<br> rt[4] := {0, 0, 0, 1, -1, 0, 0, 0, 0}<br> rt[5] := {0, 0, 0, 0, 1, -1, 0, 0, 0}<br> rt[6] := {0, 1, -1, 0, 0, 0, 0, 0, 0}<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br> http://en.wikipedia.org/wiki/E6_(mathematics)<br><br><br><br> E_7 root system<br><br> Clear[rt]<br> (*E_ 7 type Cartan matrix*)<br> r := 7<br> alp := Sum[UnitVector[r + 1, i]/2, {i, 1, 4}] -<br>   Sum[UnitVector[r + 1, i]/2, {i, 5, 8}]<br> rt[i_] :=<br>  Piecewise[{{UnitVector[r + 1, i + 2] - UnitVector[r + 1, 1 + i],<br>     i < 7}, {alp, i == 7}}, {i, 1, r}]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br>     * http://en.wikipedia.org/wiki/E7_%28mathematics%29<br><br><br><br> E_8 root system<br><br> Clear[rt]<br> (*E_ 8 type Cartan matrix*)<br> alp := -Sum[UnitVector[r, i]/2, {i, 1, r}]<br> r := 8<br> rt[i_] :=<br>  Piecewise[{{UnitVector[r, i] - UnitVector[r, 1 + i],<br>     i < 7}, {UnitVector[r, i] + UnitVector[r, i - 1], i == 7}, {alp,<br>     i == 8}}, {i, Range[r]}]<br> b[i_, j_] := (2 Dot[rt[i], rt[j]])/Dot[rt[j], rt[j]]<br> A := Table[b[i, j], {i, 1, r}, {j, 1, r}]<br> Print["root vectors"]<br> Table[rt[i], {i, 1, r}] // TableForm<br> Print["Cartan matrix"]<br> A // MatrixForm<br><br><br><br> related items<br><br>     * dilogarithm and Nahm's conjecture (mathematica)<br><br>
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==encyclopedia==
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* http://en.wikipedia.org/wiki/root_systems
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* http://en.wikipedia.org/wiki/Dynkin_diagram
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* http://mathworld.wolfram.com/DynkinDiagram.html
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[[분류:개인노트]]
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[[분류:math and physics]]
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[[분류:Lie theory]]
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[[분류:migrate]]
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==메타데이터==
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===위키데이터===
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* ID : [https://www.wikidata.org/wiki/Q534131 Q534131]
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===Spacy 패턴 목록===
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* [{'LOWER': 'root'}, {'LEMMA': 'system'}]

2021년 2월 17일 (수) 03:17 기준 최신판

encyclopedia

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'root'}, {'LEMMA': 'system'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Root_systems_and_Dynkin_diagrams(mathematica)&oldid=51954"